We study the time complexity of McKay's algorithm to compute canonical forms and automorphism groups of graphs. The algorithm is based on a type of backtrack search, and it performs pruning by discovered automorphisms and by hashing partial information of vertex labelings. In practice, the algorithm is implemented in the nauty package. We obtain colorings of Furer's graphs that allow the algorithm to compute their canonical forms in polynomial time. We then prove an exponential lower bound of the algorithm for connected 3-regular graphs of color-class size 4 using Furer's construction. We conducted experiments with nauty for these graphs. Our experimental results also indicate the same exponential lower bound.

A generalization of the random assignment problem asks the expected cost of the minimum-cost matching of cardinality k in a complete bipartire graph Kmr*, with independent random edge weights. With weights drawn from the exponential(l) distribution, the answer has been conjectured 1 to be ]1,5_>0, i+5< k (-i)('-5) ' Here, we prove the conjecture for k < 4, k = rn = 5, and k = rn = n = 6, using a structured, automated proof technique that results in proofs with relatively few cases. The method yields not only the minimum assignment cost's expectation but the Laplace transform of its distribution as well. From the Laplace transform we compute the variance in these cases, and conjecture that with k = rn = n - e<>, the variance is 2/n+ O (log n/n 2 ). We also include some asymptotic properties of the expectation and variance when k is fixed.

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The MOLGEN Chemical Identifier MOLGEN-CID is a software module freely accessible via the Internet. For a molecule or graph entered in molfile format it produces, by a canonical renumbering procedure, a canonical molfile and a unique character string that is easily compared by computer to a similar string. The mode of operation of MOLGEN-CID is detailed and visualized with examples.

We have developed an algorithm called the Universal Chemical Key (UCK) algorithm that constructs a unique key for a molecular structure. The molecular structures are represented as undirected labeled graphs with the atoms representing the vertices of the graph and the bonds representing the edges. The algorithm was tested on 236,917 compounds obtained from the National Cancer Institute (NCI) database of chemical compounds. In this paper we present the algorithm, some examples and the experimental results on the NCI database. On the NCI database, the UCK algorithm provided distinct unique keys for chemicals with different molecular structures. 1.

There is a large collection of effective algorithms for computing information about finite soluble groups. The success in computation with these groups is primarily due to a computationally convenient representation of them by means of (special forms of) power conjugate presentations. A notable omission from this collection of algorithms is an effective algorithm for computing the automorphism group of a finite soluble group. An algorithm designed for finite groups in general provides only a partial answer to this deficiency. In this thesis

. Isomorphism problems often can be solved by determining orbits of a group acting on the set of all objects to be classified. The paper centers around algorithms for this topic and shows how to base them on the same idea, the homomorphism principle. Especially it is shown that forming Sims chains, using an algorithmic version of Burnside's table of marks, computing double coset representatives, and computing Sylow subgroups of automorphism groups can be explained in this way. The exposition is based on graph theoretic concepts to give an easy explanation of data structures for group actions. 1. A General Point of View A natural goal in mathematical theories is a full description of the objects that are investigated. This goal has been successfully achieved in some cases, for example all finite abelian groups and with much more effort all finite simple groups. More often one restricted the research activity firstly to more modest problems like the pure existence of any object with som...

. In this paper we approximate large sets of univariate data by piecewise linear functions which interpolate subsets of the data, using adaptive thinning strategies. Rather than minimize the global error at each removal (AT0), we propose a much cheaper thinning strategy (AT1) which only minimizes errors locally. Interestingly, the two strategies are equivalent in all our numerical tests and we prove this to be true for convex data. We also compare with non-adaptive thinning strategies. x1. Introduction In applications such as visualization, it is often desirable to generate a hierarchy of coarser and coarser representations of a given discrete data set. Though we are primarily interested in hierarchies of scattered data sets, and in particular piecewise linear approximations over triangulations in the plane [1], we focus in this paper on univariate data sets and propose several adaptive thinning strategies. Thinning algorithms generate hierarchies of subsets by removing points ...

This paper examines the generation of Sudoku puzzles as an inverse problem, with the intention of engineering Sudoku puzzles with desired properties. We examine a number of methods that are commonly used to solve Sudoku puzzles, and then construct methods to invert each. Then, starting with a completed Sudoku board, we apply these inverse methods to construct a puzzle with a small set of clues. This is accomplished with a modified breadth-first search which ensures that our puzzle will be uniquely solvable by the methods we have described. Furthermore, this search algorithm utilizes heuristics both to reduce runtime, and to construct a Sudoku puzzle with special features. In particular we would be able to generate a puzzle of a desired